I'm not sure how difficult this would be to implement,
but it'd be amazing if the Cosmetic slots could still display while wearing a backpack from Quark!

I'm not quite sure what you want.
I just checked the Quark backpack on 1.16.5 with latest version of both mods, everything is working as intended.
What versions are you using?

I should probably explain it a little better!

Basically, Quark's backpack hides all the other inventory slots when it overrides the vanilla inventory.
If you want to equip cosmetic armor that's in your backpack for instance; you need to remove it from the backpack,
put it in your inventory/hotbar, and then re-open the cosmetic slots to equip the armor in those slots.

Sure, that's always the case with any other backpack mod... but the whole point of Quark's backpack is to
have the contents visible from within the main inventory instead of opening a separate 'bag' inventory.

Curios have the same problem; Quark's backpack changing the vanilla inventory means that even though the backpack
is 'always open', you can't access Curios or Cosmetic Armor slots unless you unequip or 'close' the backpack by opening the separate Curios/Cosmetic Armor tabs. You effectively have to open the inventory twice per slot, defeating the purpose
of having the slots integrated into the default inventory in the first place!

What I'm proposing is support for Quark's override; in short, that the Cosmetic slots would be capable of showing up
even if you have Quark's backpack open; allowing you to equip armor from your backpack and effectively 'unifying'
the inventory GUI to work the same way as it would if you were just using Cosmetic Armor with vanilla Minecraft.

You might not consider it worth the effort, and I'd completely understand that... it's probably a pretty 'niche' request.
I just find Quark's backpack UI constantly hiding my Cosmetic or Curios slots to be completely triggering! :D

Okay, I see now.
A separate integration mod that just adds an additional overriding GUI with Quark backpack opened for cosmetic slots might be more suitable in this case.
Sorry, as you said, I'm probably not going to make this integration mod for now. (I might do it in the future)
It's welcomed for anyone to make an integration mod like this.

Awesome. Thanks for reading, and a sincere thanks for all the work you've done on the mod!